Some Khovanov-Rozansky Computations

(as of December 2004)

For several reasons I believe the Khovanov-Rozansky homology theory for knots (see arXiv:math.QA/0401268) is one of the most exciting recent developments in knot theory, and very likely also out of knot theory. But I am failing miserably on understanding what it is. So just to verify that I understand the definitions correctly I wrote some programs to compute some Khovanov-Rozansky homologies.

My programs are admittedly pathetic. They are inelegant and inefficient and they hardly compute anything. Even the trefoil knot seems too large for these programs. (Though what's computed is enough to convince me that I do understand the definitions, and may already be of some very minor interest to others). Nothing was checked too carefully and all may be wrong.

My conventions follow the Khovanov-Rozansky paper and my own one page summary KRC.pdf.

The mathematica notebook KhovanovRozansky.nb (also available as KhovanovRozansky.pdf) contains pretty much all that I have computed.

If you want to re-run my computations, you need to have the packages KnotTheory`, SubQuotient.m and KRH.m on your mathematica path. Use them as in the notebook KhovanovRozansky.nb.

At the moment I have no plans of trying to improve my programs.